Journal of Knot Theory and Its Ramifications

We use Kauffman’s bracket polynomial to define a complex-valued invariant of virtual rational tangles that generalizes the well-known fraction invariant for classical rational tangles. We provide a recursive formula for computing the invariant, and use it to compute several examples.

A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case, the vector spaces agree with the state spaces of the corresponding Turaev–Viro topological quantum field theory.

In the present work, some effects of dropping the sphericality condition are investigated. In one example of nonspherical pivotal fusion categories, the string-net space counts the number of $r$-spin structures on a surface and carries an isomorphic representation of the mapping class group. Another example concerns the string-net space of a sphere with one marked point labeled by a simple object $Z$ of the Drinfeld center. This space is found to be nonzero iff $Z$ is isomorphic to a nonunit simple object determined by the nonspherical pivotal structure.

The last example mirrors the effect of deforming the stress tensor of a two-dimensional conformal field theory, such as in the topological twist of a supersymmetric theory.

We study the effects of certain local moves on Homflyptand Kauffman polynomials. We show that all Homflypt(or Kauffman) polynomials are equal in a certain nontrivial quotient of the Laurent polynomial ring. As a consequence, we discover some new properties of these invariants.

We give a description of Rasmussen’s $s$-invariant from the divisibility of Lee’s canonical class. More precisely, given any link diagram $D$, for any choice of an integral domain $R$ and a non-zero, non-invertible element $c\in R$, we define the $c$-divisibility $k_c(D)$ of Lee’s canonical class of $D$, and prove that a combination of $k_c(D)$ and some elementary properties of $D$ yields a link invariant ${\bar{s}}_c$. Each ${\bar{s}}_c$ possesses properties similar to $s$, which in particular reproves the Milnor conjecture. If we restrict to knots and take $(R,c)=(\mathbb{Q}[h],h)$, then our invariant coincides with $s$.

A virtual$n$-string is a chord diagram with $n$ core circles and a collection of arrows between core circles. We consider virtual $n$-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams.

Given a virtual 1-string $\alpha$, Turaev associated a based matrix that encodes invariants of the virtual homotopy class of $\alpha$. We generalize Turaev’s method to associate a multistring based matrix to a virtual $n$-string, addressing an open problem of Turaev and constructing similar invariants for virtual homotopy classes of virtual $n$-strings.

We construct an infinite tower of covering spaces over the configuration space of $n-1$ distinct nonzero points in the complex plane. This results in an action of the braid group ${𝔹}_n$ on the set of $n$-adic integers ${\mathbb{Z}}_n$ for all natural numbers $n\ge 2$. We study some of the properties of these actions such as continuity and transitivity. The construction of the actions involves a new way of associating to any braid $B$ an infinite sequence of braids, whose braid types are invariants of $B$. We present computations for the cases of $n=2$ and $n=3$ and use these to show an infinite family of braids close to real algebraic links, i.e. links of isolated singularities of real polynomials ${ℝ}^4\to {ℝ}^2$.

Cohomological invariants of twisted wild character varieties as constructed by Boalch and Yamakawa are derived from enumerative Calabi–Yau geometry and refined Chern–Simons invariants of torus knots. Generalizing the untwisted case, the present approach is based on a spectral correspondence for meromorphic Higgs bundles with fixed conjugacy classes at the marked points. This construction is carried out for twisted wild character varieties associated to $(\ell ,k\ell -1)$ torus knots, providing a colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.

In their paper entitled “Quantum Enhancements and Biquandle Brackets”, Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. We prove herein that if a biquandle bracket $(A,B)$ is the pointwise product of the pair of functions $(A^{\prime },B^{\prime })$ with a function $\varphi$, then $(A^{\prime },B^{\prime })$ is also a biquandle bracket if and only if $\varphi$ is a biquandle 2-cocycle (up to a constant multiple). As an application, we show that a new invariant introduced by Yang factors in this way, which allows us to show that the new invariant is in fact equivalent to the Jones polynomial on knots. Additionally, we provide a few new results about the structure of biquandle brackets and their relationship with biquandle 2-cocycles.

We show that determining the crossing number of a link is NP-hard. For some weaker notions of link equivalence, we also show NP-completeness.